Let m be the smallest odd positive iteger for which `1 + 2+ …..+` m is a square of an integer and let n be the smallest even positive integer for which `1 + 2+ ……+` n is a square of an integer. What is the value of m + n ?

Answer

To solve the problem, we need to find the smallest odd positive integer \( m \) such that the sum \( 1 + 2 + \ldots + m \) is a perfect square, and the smallest even positive integer \( n \) such that the sum \( 1 + 2 + \ldots + n \) is also a perfect square. Finally, we will calculate \( m + n \).

### Step 1: Find the smallest odd positive integer \( m \)

The sum of the first \( m \) integers is given by the formula:
\[ S_m = \frac{m(m + 1)}{2} \]
We want \( S_m \) to be a perfect square, so we need:
\[ \frac{m(m + 1)}{2} = k^2 \]
for some integer \( k \).

To find the smallest odd \( m \), we can start checking odd integers:

- For \( m = 1 \):
\[ S_1 = \frac{1(1 + 1)}{2} = \frac{1 \cdot 2}{2} = 1 = 1^2 \quad (\text{perfect square}) \]
So, \( m = 1 \).

### Step 2: Find the smallest even positive integer \( n \)

Now we need to find the smallest even positive integer \( n \) such that:
\[ S_n = \frac{n(n + 1)}{2} = k^2 \]

We can check even integers:

- For \( n = 2 \):
\[ S_2 = \frac{2(2 + 1)}{2} = \frac{2 \cdot 3}{2} = 3 \quad (\text{not a perfect square}) \]

- For \( n = 4 \):
\[ S_4 = \frac{4(4 + 1)}{2} = \frac{4 \cdot 5}{2} = 10 \quad (\text{not a perfect square}) \]

- For \( n = 6 \):
\[ S_6 = \frac{6(6 + 1)}{2} = \frac{6 \cdot 7}{2} = 21 \quad (\text{not a perfect square}) \]

- For \( n = 8 \):
\[ S_8 = \frac{8(8 + 1)}{2} = \frac{8 \cdot 9}{2} = 36 = 6^2 \quad (\text{perfect square}) \]
So, \( n = 8 \).

### Step 3: Calculate \( m + n \)

Now that we have \( m = 1 \) and \( n = 8 \), we can find:
\[ m + n = 1 + 8 = 9 \]

### Final Answer
The value of \( m + n \) is \( \boxed{9} \).